1. Decision Problems vs. Computable Problems

In Turing’s 1936 classic model, the Turing Machine is defined as a formal system that handles computable problems. It performs deterministic computation through finite steps and explicit rules. However, in his 1938 doctoral dissertation, Turing introduced a more powerful model: the Oracle Turing Machine, designed to address decision problems within formal systems that cannot be determined through internal reasoning.

We can distinguish two types of problems:

• Computable Problems: Solvable through exhaustive reasoning within a closed axiomatic system by a Turing Machine.
• Example: calculating the sum of two numbers.
• Decision Problems: Require going beyond formal systems, relying on some form of “external information” or “intuitive judgment.”
• Example: determining whether an image depicts a cat, or whether an object is an apple.

2. Perceiving Reality Is a Decision Process

The perception of the real world is, in essence, not pure computation, but a process involving ambiguity and probabilistic judgment—this closely aligns with how an Oracle Turing Machine operates:

Perceiving Reality = Decision Problem ≠ Computable Problem

We cannot rely solely on logical deduction within a formal system to determine “whether this is an apple,” because this judgment involves semantics, experience, inductive reasoning over fuzzy boundaries, and must depend on some form of intuitive decision-making mechanism.

3. Two Stages of Human Brain Operation

Take the simple act of “counting two apples” as an example. This behavior involves two levels of cognitive operations:

• Step 1: Intuitive Decision (Oracle Turing Machine): First, the brain must judge through sensory perception whether the object is an apple. This is a non-formal, intuition-dependent judgment involving ambiguity and experiential knowledge.

• Step 2: Deterministic Computation (Turing Machine): Once the object is confirmed as an “apple,” one proceeds to reason about its quantity—this is formal, deterministic, and supported by Turing Machine–like logic.
  

Therefore, the human brain can be viewed as a composite system, containing:

• Rational System (Turing Machine): Logical reasoning, rule-based computation
• Intuitive System (Oracle Machine): External perception, fuzzy judgment

4. The Key to Escaping Formal Systems: Oracle Judgment

The deductive computation within a Turing Machine is always limited by the boundaries defined by its formal system. To break through these boundaries, we must leverage the external decision capability of the Oracle Turing Machine:

From internal “computability” to external “decidability”

This is precisely what differentiates human intelligence from ordinary algorithms: Humans possess a “oracle structure” that allows them to step outside formal systems and make judgments of meaning.